A reconstruction of the state space

Now suppose that all you information you have about this system is a time series of one of the state variables, say \(x\). Because \(y(t)\) is missing we are not able to construct a state space as we saw above, but by using the result of Takens theorem we can construct a shadow manifold that recovers the information of the true state space. This reconstruction is not a copy of the original atractor, but it is a valid topological reconstruction of the original, a “shadow”.

To do this, we generate new variables by lagging the time series by multiples of a time interval \(\tau\). In this case we will use the interval between observations (\(\tau=0.2\) time units). The R commands below creates a data table object ( which is called a data frame in R) with the original time series and its lagged version. The handy function make_block by the rEDM authors does this job. Use this link to download the function to your working directory and then load it with the command:

source("../utilities/make_block.R")

And now you can create a lagged variable dataframe in R by copying the following code and pasting in the R console:

## Data frame with observations with tau = 0.2 with one lag
df1 <- make_block(data.frame(x=x1), t=time1, max_lag = 2)

A quick look at the first and last lines of the data frame can help to understand the lagged variabel structure:

## Take a look in the lagged data to understand how the function works
head(df1)

##   time         x       x_1
## 1  0.0 0.0000000        NA
## 2  0.2 0.1986693 0.0000000
## 3  0.4 0.3894183 0.1986693
## 4  0.6 0.5646425 0.3894183
## 5  0.8 0.7173561 0.5646425
## 6  1.0 0.8414710 0.7173561

tail(df1)

##       time           x         x_1
## 2496 499.0  0.49099533  0.65428133
## 2497 499.2  0.30813490  0.49099533
## 2498 499.4  0.11299011  0.30813490
## 2499 499.6 -0.08665925  0.11299011
## 2500 499.8 -0.28285377 -0.08665925
## 2501 500.0 -0.46777181 -0.28285377

By ploting the time series as a function of their lagged versions we get the shadow manifold, which is similar to the original one:

plot(x_1 ~ x, data=df1, type="l",
     xlab = "X", ylab = "X(t + tau)" )

There is a minimum number of dimensions to build a valid rceonstruction of the original atractor. In this simple case we obviously need two dimensions. Takens Theorem guarantees that the number of dimensions is up to 2D + 1, where D is the dimmensionality of the the attractor. Finding the optimal dimension for the shadow manifold is part of a vast topic called Embedology (Sauer et al. 1991). The package rEDM has some methods to find the optimal dimensions of the manifold. We will learn more in the following tutorials of this course.

Increasing the noise

Measurement errors make atractors more complex by spreading points (and thus trajectories) over the state space. Hence, the trajectories may cross in the shadow manifold, which breaks down the one to one mapping to the original atractor. Of course this effect gets larger as we increase the measurement error. We now will investigate a bit more how this affects the atractor reconstruction. The commands below doubles the measurement error of the time series and plots the shadow manifold:

x3 <- x1 + rnorm(n = length(x1), mean = 0, sd = sd(x1)/6)
df3 <- make_block(data.frame(x=x3), max_lag = 5)
plot(x_1 ~ x, data=df3, type="l", col="grey", lwd=0.5,
     ylab = "X(t + tau)", xlab = "X" )

The atractor reconstruction looks worse now, but we can overcome this problem by adding more dimensions to the shadow manifold. By doing this we unfold the shadow manifold, that is unfolding crossing trajectories. We do this adding more axes, that correspond to the time series lagged by the further time steps (\(t + \tau, \ t + 2\tau, \ t + 3\tau\)…). Even the if the original manifold has a lower dimension its is valid to reconstruct the state space with more dimensions[^1]:

You can check with the interactive plot below how an additional dimension unfolds a bit more the reconstructed atractor. In the following days we will learn some methods to find the optimal dimensions of the manifold. By now the take-home message is: the more complex the atractor is, the more dimensions you will need to have a good reconstruction.

What is the effect of \(\tau\) ?

The commands below simulate a new time series of the bimensional harmonic oscilator without measurement error but with a much smaller interval between observations (\(\tau=0.01\) time units).

## Generate the data at 0.01 time intervals
time4<-seq(0, 500, by = 0.01) 
x4<-sin(time4)
y4<-cos(time4)

This change in \(\tau\) affects the shape of the reconstruction:

df4 <- make_block(data.frame(x=x4), t=time4)
plot(x_1 ~ x, data=df4, type="l",
     xlab = "X", ylab = "X(t + tau)" )

How do you explain that? Which practical issues does this fact bring and how to solve them?

abline(0,1, col="red")

Atractor reconstruction

By using only the functions learned at this tutorial, try to reconstruct the state space of the time series] in this csv file, checking if a three-dimensional manifold improve the reconstruction. To load the data in R download the csv file and run the command:

prob2 <- read.csv("problem-time-series.csv")

And to take look in the time series plot run:

plot( X ~ time, data = prob2, type="l")

## Loads the package, if not available
library(scatterplot3d)
scatterplot3d(x = df2$x, y = df2$x_1, z = df2$x_2,
              type = "l", color="grey", lwd = 0.5,
              xlab = "x(t)", ylab = "x(t+tau)", zlab = "x(t + 2 tau)")

install.packages("scatterplot3d")